Shaft Geometry Research Articles

Finite element torsional buckling analysis of shafts with non-concentric bores

The demand for more powerful and efficient aeroengines results in the need to design improved main driveshafts. The design requirement may be to increase the torque capacity of the shafts, to reduce the outer diameter or to reduce the weight. Meeting these requirements tends to increase the likelihood of the shaft buckling and research was carried out to investigate the factors influencing the torsional buckling of shaft sections in order to provide an improved analysis method applicable to typical gas turbine aeroengine shafts. This paper uses finite element analysis to examine the effect of bore concentricity on the elastic-plastic buckling of shafts subjected to pure torsional loading. Equations were developed to predict the buckling torque for a range of shaft geometries and levels of bore eccentricity.

The effect of the orientation of the acetabular and femoral components on the range of motion of the hip at different head-neck ratios.

Prosthetic impingement due to poor positioning can limit the range of motion of the hip after total hip arthroplasty. In this study, a computer model was used to determine the effects of the positions of the acetabular and femoral components and of varying head-neck ratios on impingement and range of motion. A three-dimensional generic hip prosthesis with a hemispherical cup, a neck diameter of 12.25 millimeters, and a head size ranging from twenty-two to thirty-two millimeters was simulated on a computer. The maximum range of motion of the hip was measured, before the neck impinged on the liner of the cup, for acetabular abduction angles ranging from 35 to 55 degrees and acetabular and femoral anteversion ranging from 0 to 30 degrees. Stability of the hip was estimated as the maximum possible flexion coupled with 10 degrees of adduction and 10 degrees of internal rotation and also as the maximum possible extension coupled with 10 degrees of external rotation. The effects of prosthetic orientation on activities of daily living were analyzed as well. Acetabular abduction angles of less than 45 degrees decreased flexion and abduction of the hip, whereas higher angles decreased adduction and rotation. Femoral and acetabular anteversion increased flexion but decreased extension. Acetabular abduction angles of between 45 and 55 degrees permitted a better overall range of motion and stability when combined with appropriate acetabular and femoral anteversion. Lower head-neck ratios decreased the range of motion that was possible without prosthetic impingement. The addition of a modular sleeve that increased the diameter of the femoral neck by two millimeters decreased the range of motion by 1.5 to 8.5 degrees, depending on the direction of motion that was studied. There is a complex interplay between the angles of orientation of the femoral and acetabular components. Acetabular abduction angles between 45 and 55 degrees, when combined with appropriate acetabular and femoral anteversion, resulted in a maximum overall range of motion and stability with respect to prosthetic impingement. During total hip arthroplasty, acetabular abduction is often constrained by available bone coverage, while femoral anteversion may be dictated by the geometry of the femoral shaft. For each combination of acetabular abduction and femoral anteversion, there is an optimum range of acetabular anteversion that allows the potential for a maximum range of motion without prosthetic impingement after total hip arthroplasty. These data can be used intraoperatively to determine optimum position.

Current status and perspectives of shoulder replacement

Basis of the modern shoulder implants is the Neer II-system, a non constrained total shoulder prosthesis with conforming radii of curvature and improved protection against dislocation. The second generation of shoulder prosthesis is based on the geometric shaft design of the Neer II prosthesis and offers not only a variety of modular head- and shaft-sizes but also through different radii a physiologic rotation-translation-mechanism. The third generation of humeral head prosthesis carries the concept of an anatomic reconstruction one step further and enables the surgeon to adjust the inclination and the eccentric offset of the humeral head to restore the centre of rotation. The latest development in shoulder arthroplasty are humeral head prosthesis with a fully variable 3-dimensional modularity to independently adjust the prosthetic head position regardless of the individual shaft geometry. This achieves a 3-dimensional adaptability of the prosthetic head about the stem axis in the coronary and in the sagittal plane. Besides of the humeral shaft prosthesis an alternative concept of shoulder joint replacement is established - the replacement of the humeral head articular surface. A hemispheric surface prosthesis - cup arthroplasty - is cemented onto the residual humeral head, which eliminates the obligatory humeral head resection and the reaming of the medullary canal. Bipolar shoulder prosthesis are humeral shaft prosthesis with a bi-rotational head system. Their indication is limited to pre-existing lesions of the rotator cuff and/or the glenoid surface. The inverse total shoulder prosthesis reverses the articular surface morphology of the humeral head and the glenoid. The hemispheric glenoid component serves as the centre of rotation for the concave epiphyseal proximal humerus component. This implant is especially used in cases of massive rotator cuff deficiencies. The role of shoulder prosthesis in treating acute humeral head fractures needs special consideration. A fracture prosthesis has to restore the exact length of the humerus, the centre of rotation, and the anatomical retroversion. Positioning of the tubercula and their adequate osteosynthesis is most critical and fundamental to ensure a correct healing process. A failed consolidation of the tubercula does not lead to a satisfying result. The shoulder joint replacement can be sufficiently fixated in cemented, cementless or hybrid techniques. Today several component design variations of cemented glenoid implants exist. Their main distinction is the fixation system which can be divided into two main groups - the keel - and the peg-shaped glenoid components. The peg-shaped anchorage system shall guarantee a greater stability against shear-forces. Cementless glenoid components consist of a polyethylene inlay and a surface treated metal-back with an integrated fixation system. These fixation systems are object of intensive biomechanical research and range from conventional screw fixation to specialised cone systems and self-cutting cage-screw-systems. The critical area of cementless glenoid components is the transition zone of the PE-inlay and the metal-back because of high force development. The question of implanting a hemi- or total shoulder prosthesis is answered by the morphologic changes of the glenoid articular surface, which includes the size of the subchondral defect and the underlying etiology of the shoulder joint disease, and the age of the patient. Preoperative planning must consist of an adequate radiologic work-up - X-ray, CT or MRI - to accurately assess the glenoid morphology. G. Walch categorised the different glenoid lesions and developed a very important classification of possible glenoid deformations. To compare and evaluate the operative results one must consider the different shoulder prosthesis and the discrepancies between a hemi- and a total shoulder prosthetic replacement. Looking at the loosening and survival rate of the implant the results are

Designing the Shaft Diameter for Acceptable Levels of Stress Within an Axial-Piston Swash-Plate Type Hydrostatic Pump

In this research, the diameter of the shaft within an axial-piston swash-plate type hydrostatic pump is considered from a stress point of view. To analyze the loading of the shaft, the components within the pump are studied using a force and torque diagram and it is shown that the loads are applied differently for three main sections of the shaft. From the force and torque diagram, the actual shaft loads are determined based upon the geometry of the pump and the working pressure of the hydraulic system. Using well-accepted machine design practices, governing equations for the shaft diameter are produced for the various regions of loading along the shaft. These equations consider both bending and torsional stresses on the outer surface of the shaft. Results for the required shaft diameter are then computed for a typical pump design and compared to the geometry of an actual shaft. It is noted that stress concentrations can significantly alter these results and that the required shaft diameter can be reduced by applying the proper heat treatments and increasing the shaft strength. Finally, the designer is cautioned regarding the deflection difficulties that can arise when the shaft diameter is reduced too much. [S1050-0472(00)01704-9]

Discussion

During construction of a deep shaft at Hecla's Lucky Friday Mine, excessive anisotropic ground deformation led to significant damage to the shaft liner. This necessitated a major design change during construction, with the excavated shape of the shaft being changed from a circular to an approximately elliptical geometry. This study utilizes extensometer data collected during construction as well as a three-dimensional finite-difference model of stress redistribution around the shaft and a calibrated two-dimensional discontinuum numerical model to develop an understanding of the factors affecting the relative stability of both shaft geometries. Although the focus of the work is the deformation behavior of the foliated rock surrounding the shaft, the role of liner installation in suppressing ground displacements is also considered. The change in shaft geometry was found to be highly effective in improving stability of the shaft, with the maximum time-dependent displacement around the shaft decreasing by approximately an order of magnitude. Additionally, finite-difference models confirmed that increased stress concentrations around the shaft caused by adjacent level developments played a significant role in exacerbating ground deformation in the circular portion of the shaft.

Updated Stress Concentration Factors for Filleted Shafts in Bending and Tension

Published elastic stress concentration factors are shown to underestimate stresses in the root of a shoulder filleted shaft in bending by as much as 21 percent, and in tension by as much as forty percent. For this geometry, published charts represent only approximated stress concentration factor values, based on known solutions for similar geometries. In this study, detailed finite element analyses were performed over a wide range of filleted shaft geometries to define three useful relations for bending and tension loading: (1) revised elastic stress concentration factors, (2) revised elastic von Mises equivalent stress concentration factors and (3) the maximum stress location in the fillet. Updated results are presented in the familiar graphical form and empirical relations are fit through the curves which are suitable for use in numerical design algorithms. It is demonstrated that the first two relations reveal the full multiaxial elastic state of stress and strain at the maximum stress location. Understanding the influence of geometry on the maximum stress location can be helpful for experimental strain determination or monitoring fatigue crack nucleation. The finite element results are validated against values published in the literature for several geometries and with limited experimental data.

Modeling Rotating Shafts Using Axisymmetric Solid Finite Elements with Matrix Reduction

An axisymmetric harmonic finite element representation is used to calculate shaft lateral critical speeds and perform stability analysis. Unlike a beam element model, an axisymmetric solid element representation allows the actual rotor geometry to be modeled. A Fourier series representation allows the three-dimensional shaft geometry to be modeled in two dimensions by only considering the radial and axial coordinates. Thus, the degrees of freedom of this element type are different from the usual two translations and two rotations at each node associated with bending of a three-dimensional beam element. A required gyroscopic matrix is also presented for completeness in analysis of rotating shafts. A matrix reduction technique is used to reduce the size of the shaft mass, gyroscopic, and stiffness matrices by condensing out slave degrees of freedom in terms of the retained master degrees of freedom. The formulation is applied to various examples for verification and to investigate the effect of selection of different master degrees of freedom for this element type on the results.

Growth of shear lips in surface cracked shafts subjected to fatigue loading

Shear lip growth was monitored on the fracture surfaces of notched and plain surface cracked shafts subjected to constant amplitude axial fatigue loads. Results showed that shear lip growth in the plain specimens was slightly slower than predictions based on equations derived from through cracked plate data. This is in contrast to shear lip growth in the notched shafts, which was much slower than predicted. The reduced rate of shear lip growth in the shaft geometries is attributed to the decreasing stress intensity factor along the surface crack front and the tensile circumferential stress in the notch shafts. It was also seen that the formation of shear lips had a significant effect on the shape of the crack front. Overall specimen fatigue life, however, was relatively unaffected by shear lip growth.

Effects of age and sex on the strength and cross-sectional geometry of the femur shaft in rhesus monkeys

Effects of age and sex on the strength and cross-sectional geometry of the femur shaft in rhesus monkeys

Modelling of rotors with axisymmetric solid harmonic elements

In rotating machinery analysis, the rotating shaft is typically modeled by a series of line or beam elements. These beam elements are formulated from classical beam theory, with a basic assumption that “plane sections remain plane” during bending. However, when the rotor has abrupt changes in the diameter, this assumption is not valid; local distortion occurs near the section change, resulting in an increase in the overall bending flexibility of the shaft. In this paper the bending stiffness and natural frequencies of several shaft geometries with stepped diameter changes are compared using several modelling assumptions. The comparison includes beam approximations with both transfer matrix and finite element approaches, and available experimental data. In addition, the conceptual approach of using an axisymmetric solid model of the shaft is reviewed. Results are included for a commercial four-node element, and for a cubic element developed for this work. The effective use of matrix reduction in conjunction with an axisymmetric model is shown. The importance of non-planar distortion near section changes is reviewed.

Optimization of the Geometry of Shafts Using Boundary Elements

The problem of the optimization of the geometry of shafts is formulated in terms of boundary elements. The corresponding nonlinear programming problem is solved by Pshenichny’s Linearization method. The advantages of the boundary element method over the finite element method for optimal design of shafts are discussed, with reference to the applications.

Stability and Damped Critical Speeds of a Flexible Rotor in Fluid-Film Bearings

A method is described for calculating the threshold speed of instability and the damped critical speeds of a general flexible rotor in fluid-film journal bearings. It is analogous to the Myklestad-Prohl method for calculating critical speeds and is readily programmed for numerical computation. The rotor model can simulate any practical shaft geometry and support configuration. The bearings are represented by their linearized dynamic properties, also known as the stiffness and damping coefficients of the bearing, and the calculation includes hysteretic internal damping in the shaft and destabilizing aerodynamic forces. To demonstrate the application of the method, results are shown for an industrial, multistage compressor.

Geometry and Morphology of Crustacean Burrows in Torrey Pines and Bodega Estuaries, California: ABSTRACT

Resin casts were used to document the geometry and morphology of crustacean burrows from two California estuaries. Burrows studies include those of two ghost shrimps (Callianassa californiensis and C. longimana), a fiddler crab (Uca crenulata), and two grapsoid crabs (Pachygrapsus crassipes and Hemigrapsus oregonensis). Further documentation is under way with the use of direct observations and radiographs of ghost shrimp burrowing through layered sediment in aquaria. The Callianassa burrows are in muddy to clean sand found in the lower parts of tidal creeks and on sand flats. Their burrows have a main shaft up to 1 m long with constant diameter (up to 2 cm) except for narrowing produced by excurrent activity either at the surface or between burrow systems, and except for enlarged turn-around nodes commonly present at branches or direction changes. Up to 5 openings were observed per system; they are connected by twos and threes in horizontal to inclined Y's with the junction of the Y up to 15 cm below the surface. The geometry of the main shaft is dependent on species, intertidal position, sediment size, and layering. The burrows have a smooth internal and external morphology. The grapsoid burrows are on and above the banks of tidal creeks in slightly silty clay to slightly muddy sand. They vary from complex shapes, with several layers and entrances in a box-type framework, to a simple U-shape depending on topography, tidal level, and the number of organisms and species per system. Commonly two H. oregonensis, one P. crassipes, and one or more U. crenulata are found using parts of the same burrow system. The numerous entrances allow only lateral passage but internal enlargements permit turning around and passage of individuals. In cross section the burrows are lenticular, and the morphology of the walls is very knobby. Burrow entrances of Uca crenulata are at, or near, higher high water and extend either into a grapsoid system or a simple J-shape (up to 20 m), both of which may have a Y-shaped entrance. The entrance and extremity chamber are about twice the diameter of the knobby, cylindrical shaft which normally has a diameter less than 1 cm. End_of_Article - Last_Page 794------------

Closure to “Discussions of ‘Shaft Geometry—A Major Factor in Oil Seal Performance’” (1969, ASME J. Lubr. Technol., 91, p. 210)

Closure to “Discussions of ‘Shaft Geometry—A Major Factor in Oil Seal Performance’” (1969, ASME J. Lubr. Technol., 91, p. 210)

Shaft Geometry—A Major Factor in Oil Seal Performance

Five seal variables, viz., flex section thickness, lip length, trim diameter, material, and lip force, and two shaft variables, shaft out-of-round (OOR) and number of lobes, were investigated by means of fractional-factorial experimentation to determine the effects of shaft OOR and number of lobes on seal performance and to determine the interactions between the five seal design factors and the two shaft variables. A 1/4 replicate, 27 fractional-factorial experiment was conducted with leakage as the dependent variable to determine these effects. For the levels of OOR, lobing and shaft speed chosen, the tests indicated that shaft OOR and number of lobes in themselves had no effect. However, they appeared in significant interactions with several seal design factors. Also, at higher speeds, both OOR and lobing are extremely important. Analysis of the interactions and further test work with lobed shafts indicate that due to the interactions between several seal design factors and shaft OOR, the setting of an acceptable shaft OOR tolerance is extremely difficult. However, a general recommendation of 200 microin. maximum shaft OOR, with a minimum number of lobes, is supported by the data and is still valid.

Fatigue Strength of Turbine Shafts with Shrunk-on Discs

In 1956–57, the shafts of three large built-up steam turbine rotors developed fatigue cracks in service under very low nominal working stresses. The fatigue cracks occurred at changes in shaft section and unexpectedly large strength reduction factors were realized. The failures have been substantially explained by fatigue tests on the shaft materials, in which normal ratios of fatigue strength/ultimate tensile stress were obtained with plain specimens up to 2 5/8 in in diameter, and strength reduction factors up to 10 were obtained with model rotor shafts about 4 in in diameter with shrunk-on collars. Additional tests have verified that modifications introduced into service at the time of the failures were effective in substantially increasing fatigue strength, and have shown the effect on fatigue strength of variations in the geometry of shafts and wheel bores.